B.1 Bloch Wave Functions Prefactors

To obtain CA and CB in Eq. 3.18, one can substitute Eq. 3.6 and Eq. 3.19 into the Shršodinger equation H|ψ= E|ψ. Considering an A-type carbon atom at some atomic site n, and its three nearest neighbors, the Hamiltonian can be written as:

H  =  t|B       ⟩⟨A  | + t|B′     ⟩⟨A  | + t|B     ⟩⟨A |.
         N-n+1    n       N -n+1    n       N- n   n
(B.1)

Using Eq. B.1 along with the wave functions obtained in Eq. 4.12, one obtains:

         A                   B
ECAeikxx n sin(nθ) = tCBeikxx N-n+1 sin((N - n + 1 )θ )
                                ′
                     + tC  eikxxBN- n+1 sin ((N  - n + 1)θ)
                          B
                            ikxxBN- n
                     + tCBe        sin((N - n )θ).
(B.2)

Therefore, the relation between CA and CB can be written as:

                   [ (                                 )
                                         ik (xB′   -xA )
ECA  sin(nθ) = tCB     eikx(xBN-n+1-xAn) + e x  N-n+1  n

                                                               ]
                                    ikx(xBN -n-xAn)
               sin ((N - n + 1 )θ ) + e           sin((N  - n)θ)
                   [      ( √ --    )                                   ]
                              3
             = tCB   2cos   ---kxacc  sin ((N  - n + 1)θ) + sin((N - n )θ) .
                             2
(B.3)

By employing the relation sin(x) sin(y) = (12)[cos(x-y) - cos(x + y)] and using Eq. 4.22,

EC    = - tC  ----sin(θ)----.
    A       B sin((N  + 1)θ)
(B.4)

Analogously, for the N - n + 1th B-type carbon atom one can obtain the following relation:

                              [     (         )                        ]
                                      √3--
ECB  sin ((N -  n + 1)θ) = tCA  2 cos  ----kxacc  sin(n θ) + sin((n - 1)θ)
                                       2
(B.5)

which gives

EC    = - tC  ----sin(θ)----.
    B       A sin((N  + 1)θ)
(B.6)

From Eq. B.4 and Eq. B.6, one can find that CA = ±CB.
Also, the dispersion relation can be found by multiplying Eq. B.3 by Eq. B.5,

  2
E  CACB  sin(nθ)sin((N -  n + 1)θ)(         )
                                   √ --
                 = t2CACB  [4 cos2  --3kxacc  sin((N -  n + 1)θ)sin(nθ)
                                    2
                          ( √ --    )
                            --3-
                   + 2cos    2 kxacc  sin((N - n +  1)θ)sin ((n -  1)θ)
                          ( √ --    )
                              3
                   + 2cos   ---kxacc  sin((N - n )θ)sin(nθ)
                             2
                   + sin ((N  - n )θ )sin ((n - 1 )θ )].
(B.7)

With the help of trigonometric identities and Eq. 4.22, this expression can be reformatted as

        [          ( √ --    )         ( √ --    )       ]1∕2
                  2    3                   3
E  = ±t  1 + 4 cos   ---kxacc   + 4cos   ---kxacc  cos(θ)    .
                      2                   2
(B.8)